Tight-binding description of the quasiparticle dispersion of graphite and few-layer graphene

A universal set of third-nearest-neighbor tight-binding (TB) parameters is presented for calculation of the quasiparticle (QP) dispersion of $N$ stacked $s{p}^2$ graphene layers $\left(N=1\dots \infty \right)$ with $AB$ stacking sequence. The present TB parameters are fit to ab initio calculations on the GW level and are universal, allowing to describe the whole $\pi$ “experimental” band structure with one set of parameters. This is important for describing both low-energy electronic transport and high-energy optical properties of graphene layers. The QP bands are strongly renormalized by electron-electron interactions, which results in a 20% increase in the nearest-neighbor in-plane and out-of-plane TB parameters when compared to band structure from density-functional theory. With the new set of TB parameters we determine the Fermi surface and evaluate exciton energies, charge carrier plasmon frequencies, and the conductivities which are relevant for recent angle-resolved photoemission, optical, electron energy loss, and transport measurements. A comparision of these quantitities to experiments yields an excellent agreement. Furthermore we discuss the transition from few-layer graphene to graphite and a semimetal to metal transition in a TB framework.

Figure 1

(Color online) (a) The graphite unit cell consists of four atoms denoted by $A_1$, $B_1$, $A_2$, and $B_2$ (light blue). The red arrows denote the interatomic tight-binding hopping matrix elements ${\gamma }_0^1,{\gamma }_0^2,{\gamma }_0^3,{\gamma }_1,\dots ,{\gamma }_5$. The overlap matrix elements $s_0^1$, $s_0^2$, and $s_0^3$ (not shown) couple the same atoms as ${\gamma }_0^1$, ${\gamma }_0^2$, and ${\gamma }_0^3$. (b) The 3D Brillouin zone of graphite with the high-symmetry points and the coordinate system used throughout this work.

Figure 2

The TB fit along $k_z$ direction at $k_x=k_y=0$ ($KH$ axis) for (a) LDA and (b) $GW$. ◻ denotes LDA calculations and △ denotes $GW$ calculations taken from Ref. 13 that were used for the fitting.

Figure 3

Upper panel: The TB fits along $k_y$ direction at $k_z=0$ ($K$ point) for (a) LDA and (b) $GW$. Lower panel: the $k_y$ dispersion at $k_z=0.47{\text{Å}}^{-1}$ ($H$ point) for (c) LDA and (d) $GW$. ◻ denotes LDA calculations and △ denotes $GW$ calculations that were used for the fitting (taken from Ref. 13).

Figure 4

The TB fits of equienergy contours at $k_z=0$ for (a) TB-LDA and (b) $\text{TB-}GW$.

Figure 5

The in-plane quasiparticle dispersion for (a) $k_z=0$ and (b) $k_z=0.47{\text{Å}}^{-1}$ calculated by $\text{TB-}GW$. The symbols ${\pi }_1-{\pi }_4$ denote the four $\pi$ bands of graphite.

Figure 6

(a) The weakly dispersive band that is responsible for the formation of electron and hole pockets. △ denotes $GW$ calculations (with the impurity doping level adjusted to reproduce the experimental gap at $H$ point) and the line the $\text{TB-}GW$ fits. (b) shows a magnification of the minority hole pocket that is caused by the steeply dispersive band close to $H$.

Figure 7

(Color online) $\text{TB-}GW$ QP band structure around [(a), (b)] $K$ point and [(c), (d)] $H$ point. The points 1, 2, and 3 denote the dispersion we use to determine electron and hole masses (see Fig. 9). The points 1, 2, and 3 span a parabola. In [(a), (b)] the parabola is along the $KM$ direction with heaviest electron masses and in [(c), (d)] the hole masses are isotropic around $H$.

Figure 8

(Color online) Evaluation of the in-plane (a) electron and (b) hole masses around $K$ and $H$, respectively. The dispersions along the parabolas depicted by points 1-2-3 in Fig. 7 are taken for evaluation of the masses. The effective masses perpendicular to the layers in $z$ direction are shown in (c) for electrons and in (d) for holes. ◻ correspond to $\text{TB-}GW$ values and the lines are parabolic fits.

Figure 9

The $k_z$ dependence of the in-plane electron mass $\left(m_e^{\ast }\right)$ and hole mass $\left(m_h^{\ast }\right)$ calculated by $\text{TB-}GW$. The masses are evaluated along the parabolas as shown in Fig. 8.

Figure 10

(Color online) Doping dependence of (a) the electron and hole carrier density, (b) the number of electrons and holes, and (c) the stochiometric dopant to carbon ratio. The red (green) lines represent electron (hole) carriers.

Figure 11

(Color online) The Fermi surface of doped graphite in the dilute limit for electron doping and hole doping. The red (green) surfaces are the electron (hole) pockets of the Fermi surface. It is clear that at $\pm 25\text{meV}$ we have a semimetal to metal transition.

Figure 12

Quasiparticle $\pi$ bands for [(a), (b)] bilayer, [(c), (d)] trilayer, and [(e), (f)] quadlayer graphene. The right panels show a magnification of the band structure around $K$ point. These band structures have been calculated by 3NN $\text{TB-}GW$ using the parameters from Table . The corresponding Hamiltonian and overlap matrices are given in Appendix.

Figure 13

(Color online) Evolution of the eigenvalue spectrum for FLG from $N=1\dots 30$ graphene layers calculated by $\text{TB-}GW$ at the $K$ point. The dashed lines labeled by ${\pi }_1$ and ${\pi }_4$ denote the lower and upper limits of the total bandwidth at $K$ point in 3D graphite. A pattern that connects the first, second, etc. energies of FLG is emerging (see text).